Nature and Science, 1(1), 2003, Huang, Liang and Liu, Applying of RAGA
Applying on RAGA in Estimating Internal Rate
of Return of Hydropower Project
Xiaorong Huang*, Chuan Liang*, Zhiyong Liu**
* School of Hydropower Engineering, Sichuan University, Chengdu, Sichuan 610065, China,
Hxiaorong@163.net; ** School of Network, Sichuan University, Chengdu, Sichuan 610065, China
Abstract: Internal rate of return, a kind of important economic analysis index, is often used by evaluating
economic feasibility for engineering. Substantively, estimating internal rate of return is to solve higher-order
equation with one unknown. Due to its precision disadvantage, complicated calculation, and especially
theories lacks, traditional linear interpolation is unsuited to do with these questions. Real Coding
Accelerating Genetic Algorithm (RAGA), a new method, has overall optimization ability, less calculation
quantity, adaptability to interspace size of the optimistic search variety, swiftly calculation and high
accuracy. It can be used and proved to calculate internal rate of return effectually. [Nature and Science
2003;1(1):72-74].
Key words: RAGA; internal rate of return; linear interpolation
Introduction
The investment of hydropower engineering was
gradually taken back by its net income of each year. If
everything is OK, at the end of economic life, the
amount of the investment can be returned at the right
moment. Because this process has nothing to do with
outside factors, only with its own investment, net
income of each year and increment rate of possessive
funds, the rate of discount is so called as internal rate
of return. Solving higher-order equation with one
unknown usually can get internal rate of return, an
unknown rate. Unfortunately, at present, a simple and
effectual solution method cannot be really obtained
because linear interpolation, which is often used
before, can’t attain satisfactory accuracy for that there
is no linear relation between rate of discount and
internal rate of return (Liu, 2003). Real Coding
Accelerating Genetic Algorithm (RAGA) can resolve
this problem nicely, and it can acquire desired result
from big scope, meanwhile its accuracy is obviously
higher than what traditional linear interpolation can do.
Moreover, RAGA does not need attempt to calculate it
many times, then the calculation is reduced greatly,
and also overcome the result which varied with
different persons only because this linear interpolation
method is adopted. As we know, Internal rate of return
can make sure the capital income and foreshow the
biggest interest rate afforded without financial crisis.
Hydropower engineering usually possessing big
project investment, a long project constructing and
project repayment period, needs higher accuracy that
can reflect the income ability of investment well. In
this way, RAGA provides the new method for us with
the way of thinking. Scientific and dependable data
can be available.
1. The Brief Introduction of RAGA
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Standard Genetic Algorithm (SGA), the traditional
genetic algorithm, usually adopts the coding method of
binary system. Its individual genotype is also called
binary coded string that only uses coding sign 0 and 1.
Binary coding is not only simple and easy to proceed its
crossover, selection and mutation etc., but also easy to
make good use of the mode axioms to theoretically
analyze algorithm. In despite of this, Standard Genetic
Algorithm’s shortage is still obvious: it can’t reflect the
special structure characteristics of a problem enough,
and its part-search ability is also worse due to its
random attributes when optimization question of
continuous functions is handled. Furthermore, to
research optimization question of continuous functions
containing many dimensions and needing higher
accuracy, the binary coding shows disadvantages.
Firstly, the binary coding exits mapping error when
continuous functions proceed to discrete process. If the
length of coding string is shorter, the requested accuracy
cannot be obtained. If the length of coding string is
longer, even though its accuracy is improved, the search
scope of genetic algorithm swiftly extends. Secondly,
the binary coding cannot reflect particular attributes of
problems well. It is difficult to generate special genetic
operator for special problems. As a result, many
precious experiences from research on classic optimal
algorithm cannot be made good usage, and special
restricted condition isn’t easy to be handled. In order to
overcome these shortages of binary coding method, we
may adopt real number coding, and make the individual
coding length fitly equal to decision variable number.
This kind of coding has a few advantages: (1) be
qualified to denote bigger number; (2) be qualified to
obtain higher accuracy; (3) be qualified to search from
big scope; (4) reduce the calculation complexity and
improve calculation efficiency; (5) be easy to unite
genetic algorithm and classic optimal algorithms; (6) be
easy to design genetic operators aiming to specialized
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Nature and Science, 1(1), 2003, Huang, Liang and Liu, Applying of RAGA
problems; (7) be easy to handle complicated restricted
conditions.
RAGA can gradually narrow the scope of optimistic
variable by making use of accelerating circulation. The
accuracy of solution is improved quickly.
RAGA’s control parameters are made up of
community scale n , excellent individual number s ,
mutation probability Pm , crossover probability Pc ,
and their value will influence final optimistic value.
Having done a great deal of numerical experiments and
actual application, we obtain experimental relation
between n and s :
2. The Estimation of Internal Rate of Return
Internal rate of return is kinds of discount rate, which
can make total amount of net present value of each
year in project economic life, happen to be equal to
zero. We now introduce the expression below:
n
 (CI
t 0
(1)
 CO) t (1  I R R
) t  0
Equation inside: IRR——internal rate of return;
s / n  6%
(CI  CO) t ——net present value of T year
（4）
When we apply RAGA, we suggest n : over 300
number, s : over 20, Pm : 0.8, Pc : 0.8 (Jin, 2000). If
these are satisfied, we can get better result.
n ——life period of project
As we know, equation (1) is a higher-order equation
with one unknown. Hence, it isn’t easy to be solved.
At present, the linear interpolation method is usually
used. Its expression is as the following:
Figure 1. Flow Chart for RAGA
Presetting Parameters of AGA
NPV
1
I R R i1 
(i2  i1 ) （2）
NPV

N
P V2
1
Production of Forerumner
Individuals
i1 ——lower rate of return that we attempt to
calculate；
i 2 ——higher rate of return that we attempt to
Decoding and Evaluation
calculate；
——net present value we calculate by i1
NPV
1
(positive value);
——net present value we calculate by i 2
NPV
2
(negative value).
Because hydropower engineering’s lifetime is long
(several decades), it is difficult to calculate internal
rate of return by traditional mathematics method.
According to the equations above, we consider that
the accuracy of the linear interpolation method is low
and its calculation is complicated. Hence, we attempt
to turn equation （1 ） to minimization problem,
another method to calculate internal rate of return.
From equation（2－1）, they obtain
m i nQ  min
n
 (CI  CO)
t 0
t
Selection, Crossover,mutation
Iteration twice
Whether or Not
not
Yes
Output
(1  I R R) t （3）
4. Example Analysis
Zipingpu water conservancy project, located at 6 km
and on the upstream from Dujiangyan, Chengdu, China,
is a key project to exploit the upstream of Minjing River
reasonably. It tackles the difficulties of water and power
shortage and flood control in Dujianyan irrigation area
and Chengdu city. Its main functions are irrigation and
municipal water supply, comprehensive benefits of
generation, flood control, tourism and aquaculture. Its
installed capacity has 0.76 million KW, average annual
energy output has 3.417 billion KW.h. The total static
investment reaches 5.75039 billion yuan (Chinese dollar)
Therefore, internal rate of return is really to solve
non-linear optimization problem, and RAGA is easy
to handle it.
3. The Process That Model Realizes
Figure 1 shows the steps and calculation process of
RAGA. In comparison, selection, crossover and
mutation of RAGA proceed to parallel computation. In
consequence, RAGA’s search scope is bigger than
SGA’s, and there are more chances to get optimal value.
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Nature and Science, 1(1), 2003, Huang, Liang and Liu, Applying of RAGA
and the total dynamic investment reaches 6.23600 of each year is shown in Table 1, and the calculation of
billion yuan. Its construction period will last 7 years and the internal rate of return of Zipingpu engineering by
operation period will last 30 years, so the total economic different measures according to equation (3) is shown in
calculation period will last 37 years. The net cash flow Table 2.
Table 1. Zipingpu Project Net Cash Flow (unit: ten thousand Chinese yuan)
Year
NCF
Year
NCF
Year
NCF
Year
NCF
Year
NCF
1
–26274
9
59031
17
73059
25
72553
33
56628
2
–63040
10
76491
18
72996
26
72489
34
56628
3
–96349
11
76047
19
72932
27
72426
35
56628
4
–116874
12
75575
20
72869
28
56881
36
–263074
5
–122121
13
75010
21
72806
29
56818
6
–84969
14
74415
22
72742
30
56754
7
–28916
15
73786
23
72679
31
56691
8
53465
16
73122
24
72616
32
56628
Table 2. The Internal Rate of Return of Zipingpu Engineering (unit: ten thousand Chinese yuan)
method
result
Internal rate of return
Remains value Q
8.81%
8.79%
8.7870%
8.7880%
8.7878%
997.945
97.021
38.4226
6.7341
2.2189
Linear interpolation
Fminbnd function
RAGA（accelerate once）
RAGA（accelerate two times ）
RAGA（accelerate three times）
It must be pointed out that the RAGA method is
applied straight when the mark of cash flow change
once. If it changes many times, all this shows that the
income from the engineering investment is laid out
some other external engineering. The external rate of
return isn’t always as same as the internal rate of
return. If so, firstly we should convert the surplus
income into external investment according to the
external rate of return, then this part fund is further
returned to internal engineering. The literatures
suggest that we may estimate the internal rate of
return by net investment examination method to any
cash flow, whose expression would be of the form:
m i nQ  NI n (r )
Finally, we may apply the RAGA method to proceed
the calculation.
5. Conclusion
Estimating the internal rate of return is actually to solve
a kind of complicated and non-linear problem, by the
RAGA method, higher accuracy, less error, more
rationality we can obtain. When the cash flow mark of
engineering change over twice, we should avoid
confusing external rate of return and internal rate of
return, otherwise it will bring important error for
decision-making.
a 0 , a1 , a 2 , , a n , r means internal rate of
return, e means external rate of return, so the net
investment of j year can be written in the form:
NI 0  a0
Correspondence to:
Xiaorong Huang
School of Hydropower Engineering, Sichuan University
Chengdu, Sichuan 610065, China
Telephone: 01186-28-8540-5610
Cellular phone: 01186-13608058560
E-mail: Hxiaorong@163.net
(5)
NI j  N j 1 (1  i )  a j , ( j  1,2, , n)
(6)
If NI j 1  0 j  1,2,  , n  , the parameter i is
References
Jin J, Ding J. Genetic Algorithm and Its Applications to Water
Science. Publishing House of Sichuan University. Chengdu, China.
2000;112-33.
Liu X. Engineering Economy Analysis. Publishing House of Xinan
Jiaotong University. Xian, China. 2003;71-4.
Science and Technology Product Development Centre of Fei Si.
The Assistant Optimizing Calculation and Design of the MATHLAB
6.5. Publishing House of Electronics Industry. Beijing, China.
2003;41-2.
set by internal rate of return. If not, the parameter i
is set by external rate of return. At the same time, the
net investment of n year must be equal to zero. We
now introduce the expressions below:
NI n  NI n (r )  0
(7)
Furthermore, we obtain
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